L2/11‐434 Technical Reports Proposed Update Unicode Technical Report #25 UNICODE SUPPORT FOR MATHEMATICS Authors Barbara Beeton [email protected], Asmus Freytag [email protected], Murray Sargent III [email protected] Date 2011‐11‐02 This Version http://www.unicode.org/reports/tr25/tr25‐13.pdf Previous Version http://www.unicode.org/reports/tr25/tr25‐12.pdf Latest Version http://www.unicode.org/reports/tr25 Source Document http://www.unicode.org/reports/tr25/tr25‐13.docx Revision 13 Summary The Unicode Standard includes virtually all standard characters used in mathematics. This set supports a wide variety of math usage on computers, including in document presentation lan‐ guages like TEX, in math markup languages like MathML and OpenMath, in internal represen‐ tations of mathematics for applications like Mathematica, Maple, and MathCAD, in computer programs, and in plain text. This technical report describes the Unicode support for mathe‐ matics and gives some of the imputed default math properties for Unicode characters. Status A Unicode Technical Report (UTR) contains informative material. Conformance to the Standard does not imply conformance to any UTR. Other specifications, however, are free to make normative references to a UTR. Please submit corrigenda and other comments with the online reporting form [Feedback]. Re‐ lated information that is useful in understanding this document is found in the References. For the latest version of the Unicode Standard see [Unicode]. For a list of current Unicode Tech‐ nical Reports see [Reports]. For more information about versions of the Unicode Standard, see [Versions]. Unicode Technical Report #25 1 Unicode Support for Mathematics Contents UNICODE SUPPORT FOR MATHEMATICS .................................................................................................................. 1 1. OVERVIEW............................................................................................................................................................................................... 3 2. MATHEMATICAL CHARACTER REPERTOIRE ......................................................................................................................... 4 2.1 Mathematical Alphanumeric Symbols Block ........................................................................................................... 4 2.2 Mathematical Alphabets ......................................................................................................................................... 5 2.3 Fonts Used for Mathematical Alphabets ................................................................................................................. 8 2.3.1 Representative Glyphs for Greek Phi ............................................................................................................. 10 2.3.2 Representative Glyphs for U+2278 and U+2279 ............................................................................................ 10 2.4 Locating Mathematical Characters ........................................................................................................................ 11 2.5 Duplicated Characters ........................................................................................................................................... 11 2.6 Accented Characters .............................................................................................................................................. 12 2.7 Operators ............................................................................................................................................................... 13 2.8 Superscripts and Subscripts ................................................................................................................................... 15 2.9 Arrows ................................................................................................................................................................... 15 2.10 Delimiters ............................................................................................................................................................ 16 2.11 Geometrical Shapes ............................................................................................................................................. 19 2.12 Other Symbols ..................................................................................................................................................... 22 2.13 Symbol Pieces ...................................................................................................................................................... 22 2.14 Invisible Operators ............................................................................................................................................... 23 2.15 Fraction Slash and Other Diagonals ..................................................................................................................... 24 2.16 Other Characters ................................................................................................................................................. 26 2.17 Negations ............................................................................................................................................................. 26 2.18 Variation Selector ................................................................................................................................................ 28 2.19 Novel Symbols not yet in Unicode ....................................................................................................................... 29 3. MATHEMATICAL CHARACTER PROPERTIES ........................................................................................................................ 30 3.1 Classification by Degree of Mathematical Usage ........................................................................................... 30 3.1.1 Strongly Mathematical Characters ................................................................................................................. 30 3.1.2 Weakly Mathematical Characters .................................................................................................................. 31 3.1.3 Other .............................................................................................................................................................. 33 3.2 Classification by Typographical Behavior ....................................................................................................... 33 3.2.1 Alphabetic ..................................................................................................................................................... 33 3.2.2 Operators ...................................................................................................................................................... 33 3.2.3 Large Operators .............................................................................................................................................. 34 3.2.4 Digits .............................................................................................................................................................. 34 3.2.5 Delimiters ...................................................................................................................................................... 34 3.2.6 Fences ............................................................................................................................................................ 34 3.2.7 Combining Marks........................................................................................................................................... 35 4. IMPLEMENTATION GUIDELINES ................................................................................................................................................ 35 4.1 Use of Normalization with Mathematical Text ...................................................................................................... 35 4.2 Bidirectional Layout of Mathematical Text ........................................................................................................... 36 4.3 Input of Mathematical and Other Unicode Characters ......................................................................................... 37 4.4 Use of Math Characters in Computer Programs .................................................................................................... 38 4.5 Recognizing Mathematical Expressions ................................................................................................................. 39 4.6 Some Examples of Mathematical Notation ........................................................................................................... 40 5. DATA FILES ........................................................................................................................................................................................... 41 5.1 Mathematical Classification ........................................................................................................................... 41 5.2 Mapping to other Standards .......................................................................................................................... 42 6. SECURITY CONSIDERATIONS ......................................................................................................................................................